Method and apparatus for equal energy codebooks for coupled antennas with transmission lines

ABSTRACT

A method and apparatus provide equal energy codebooks for coupled antennas with transmission lines. A plurality of precoders can be received from a codebook in a transmitter having an antenna array. Each precoder of the plurality of precoders can be transformed to a transformed precoder such that the transmit power for each transformed precoder is equal to the transmit power for each of other transformed precoders of the plurality of precoders. The transmit power can be expressed as a quadratic form with respect to the corresponding precoder. The quadratic form can be based on a transmission line impedance of a transmission line between a signal source and the antenna array. A signal can be received from the signal source. A transformed precoder of the plurality of transformed precoders can be applied to the signal to generate a precoded signal for transmission over a physical channel. The precoded signal can be transmitted.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is related to an application entitled “Method andApparatus for Optimizing Antenna Precoder Selection with CoupledAntennas,” U.S. application Ser. No. 15/016,140, Motorola Mobilitydocket number MM01141, filed on Feb. 4, 2016, an application entitled“Method and Apparatus for Equal Energy Codebooks for Antenna Arrays withMutual Coupling,” Motorola Mobility docket number MM01601, U.S.application Ser. No. 14/855,693, filed on Sep. 16, 2015, and anapplication entitled “Method and Apparatus for Equal Energy Codebooksfor Antenna Arrays with Mutual Coupling,” Motorola Mobility docketnumber MM02071-US-NP, U.S. application Ser. No. 15/157,754, filed on May18, 2016, all commonly assigned to the assignee of the presentapplication, which are hereby incorporated by reference.

BACKGROUND 1. Field

The present disclosure is directed to a method and apparatus for equalenergy codebooks for coupled antennas with transmission lines.

2. Introduction

Presently, wireless communication devices communicate with othercommunication devices using wireless signals. Many wirelesscommunication devices have multiple antennas that can transmit morefocused signals to a receiving device using antenna beamforming.Multiple telecommunication standards define antenna precoder codebooksto support antenna beamforming and or Multiple-Input/Multiple Output(MIMO) transmission with feedback from the receiver. Telecommunicationstandards that employ codebooks of precoders include the ThirdGeneration Partnership Project High Speed Packet Access (3GPP HSPA) andLong Term Evolution (LTE) standards, the IEEE 802.11 and 802.16standards. In all of these standards, the precoders that are definedhave the property that each precoding vector has equal L2 norm with theassumption that the precoders are applied to the antenna array in such away that precoders having equal L2 norm yield antenna patterns withequal power in the far field.

In each of the above telecommunication standards, precoders are used incombination with reference symbol transmissions from a transmitter sothat a receiver can evaluate the channel that would result fromapplication of each of the precoders. The receiver applies each of theprecoders to the reference symbols in order to evaluate the channelquality. It then signals the index of the best precoder and thecorresponding channel quality back to the transmitter. For sometransmission modes, the precoder used for the data transmission issignaled to the receiver, which then applies the precoder to estimatethe channel for the data symbols.

Implicit in the operation of these types of systems is the assumptionthat the precoders are applied in a manner such that the antenna patterncorresponding to each precoder has the same transmit power. The reasonfor this assumption is that it is the objective of the receiver toselect the precoder which maximizes its channel quality, and thus theachievable data rate, for a given transmit power. In the case of asingle user, this will maximize the transmission range of a fixed datarate, or alternatively, the achievable data rate at a fixed range.Alternatively, for multi-user systems, it is desirable to minimize thetransmit power needed to achieve a given data rate for each user, asthis transmit power is interference for all users other than the targetuser.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to describe the manner in which advantages and features of thedisclosure can be obtained, a description of the disclosure is renderedby reference to specific embodiments thereof which are illustrated inthe appended drawings. These drawings depict only example embodiments ofthe disclosure and are not therefore to be considered to be limiting ofits scope. The drawings may have been simplified for clarity and are notnecessarily drawn to scale.

FIG. 1 is an example block diagram of a system according to a possibleembodiment;

FIG. 2 is an example illustration of a Thevenin source and two-portantenna array model with a transmission line according to a possibleembodiment;

FIG. 3 is an example illustration of a matching network between aThevenin source and an antenna array with an impedance matrix accordingto a possible embodiment;

FIG. 4 is an example illustration of a matching network combined with anantenna array with a resulting combined impedance matrix according to apossible embodiment;

FIG. 5 is an example illustration of variation of transmit power as afunction of phase for a two-element dipole array with half-wavelengthspacing driven by a 50 ohm Thevenin source and 50 ohm transmission linesaccording to a possible embodiment;

FIG. 6 is an example illustration of a Norton source and two-portantenna model with a transmission line according to a possibleembodiment;

FIG. 7 is an example illustration of a matching network between a Nortonsource and an antenna array with an impedance matrix according to apossible embodiment;

FIG. 8 is an example illustration of a matching network combined with anantenna array with a resulting combined impedance matrix according to apossible embodiment;

FIG. 9 is an example graph of variation of transmit power as a functionof phase for a two-element dipole array with half-wavelength spacingdriven by a 50 ohm Thevenin source with isolators and 50 ohmtransmission lines according to a possible embodiment;

FIG. 10 is an example flowchart illustrating the operation of a wirelesscommunication device according to a possible embodiment;

FIG. 11 is an example flowchart illustrating the operation of a wirelesscommunication device according to a possible embodiment; and

FIG. 12 is an example block diagram of an apparatus according to apossible embodiment.

DETAILED DESCRIPTION

Embodiments provide a method and apparatus for equal energy codebooksfor coupled antennas with transmission lines. According to a possibleembodiment, a plurality of precoders can be received from a codebook ina transmitter having an antenna array. Each precoder of the plurality ofprecoders can be transformed to a transformed precoder such that thetransmit power for each transformed precoder is equal to the transmitpower for each of the other transformed precoders of the plurality ofprecoders. The transmit power can be expressed as a quadratic form withrespect to the corresponding precoder. The quadratic form can be basedon a transmission line impedance of a transmission line between a signalsource and the antenna array. A signal can be received from the signalsource. A transformed precoder of the plurality of transformed precoderscan be applied to the signal to generate a precoded signal fortransmission over a physical channel. The precoded signal can betransmitted.

FIG. 1 is an example block diagram of a system 100 according to apossible embodiment. The system 100 can include a transmitting device110 and a receiving device 120. The transmitting device 110 can be aUser Equipment (UE), a base station, or any other device that cantransmit wireless signals. Similarly, the receiving device 120 can be aUE, a base station, or any other device that can receive wirelesssignals. A UE can be a wireless terminal, a portable wirelesscommunication device, a smartphone, a cellular telephone, a flip phone,a personal digital assistant, a device having a subscriber identitymodule, a personal computer, a selective call receiver, a tabletcomputer, a laptop computer, or any other device that is capable ofsending and receiving wireless communication signals. A base station canbe a wireless wide area network base station, a wireless local areanetwork base station, an enhanced NodeB (eNB), an access point, or anyother base station.

The transmitting device 110 can include a precoder transformationcontroller 112, a codebook 114, and an antenna array 116. The precodertransformation controller 112 can be one element or can be distributedbetween different elements. For example, the precoder transformationcontroller 112 can be part of a processor, can be part of a transceiver,can be part of a precoder, can be part of other elements in atransmitting device, and/or can be distributed between combinations ofelements in a transmitting device and/or over cloud computing. Thereceiving device 120 can include at least one antenna 122. For example,in some embodiments the receiving device 120 can have one antenna and inother embodiments the receiving device 120 can have an array ofantennas. The transmitting device 110 can also act as a receiving deviceand the receiving device 120 can also act as a transmitting devicedepending on which device is currently transmitting or receiving.

If there is no mutual coupling of the antenna array 116, such as atransmit array, then it will be true that antenna precoding vectorshaving equal L2 norm will yield antenna patterns with equal power (notethat some assumptions may be necessary, such as the antennas havingequal self-impedance). However, if the antennas of the antenna array 116are coupled, then the antenna patterns resulting from two precodershaving the same L2 norm can differ in transmit power by several dB. Theamount of this difference can depend on multiple factors, including themutual coupling coefficients, the type of source used to drive thearray, the source impedance, and/or other factors. The power deliveredto the antenna array 116 can depend on relative phases of the inputs ofa vector voltage source or a vector current source, on the sourceimpedances, and/or on other factors.

In the case that a transmission line is used between a source of asource signal 118 and the antenna array 116, the transmitted power canvary with the relative phase of the input voltage vector or input sourcevector, and this variation can further depend on the source impedance,the transmission line impedance, the length of the transmission line,the antenna impedance (and any matching circuitry), and/or otherfactors. If a transmission line is used between the source and theantenna array 116 in combination with an isolator at the source, thetransmitted power can still vary with the relative phase of the inputvoltage vector or input source vector, but this variation may now dependonly on the transmission line impedance, the antenna impedance (and anymatching circuitry), and/or other factors so that the power variationmay no longer depend on the source impedance or the length of thetransmission line.

Embodiments can show that the power variation as a function of theprecoder can be expressed as a quadratic form, which is non-negativedefinite for the cases in which transmission lines are used between asource and an antenna array. Using these quadratic forms for thetransmit power, at least two methods can be used for mapping theprecoders to antenna patterns with equal transmit power. In a firstmethod, each precoder can be scaled by the inverse square root of thetransmit power that results from the unscaled precoder. In the secondmethod, the set of precoders can be transformed by multiplying eachprecoder by a matrix, so that the resulting set of precoders all map toantenna patterns having the same power. If precoder-based channelestimation is used in combination with common reference symbols, thenthe same precoder transformation can also be applied to the commonreference symbol precoders.

FIG. 2 is an example illustration 200 of a Thevenin source and two-portantenna array model with a transmission line according to a possibleembodiment. FIG. 3 is an example illustration 300 of a matching networkexplicitly shown between a Thevenin source and an antenna array withimpedance matrix Z according to a possible embodiment. FIG. 4 is anexample illustration 400 of a matching network combined with an antennaarray with a resulting combined impedance matrix Z′ according to apossible embodiment, where the combined impedance matrix Z′ can be usedinstead of the impedance matrix Z in equations herein to account for amatching network.

This embodiment can consider transmit power with a Thevenin Source,transmission lines, and no isolators at the source. For example, theillustration 200 shows an antenna array 230 driven by a Thevenin source210 with a transmission line 220 between the source 210 and the antennaarray 230. This can be used to determine transmit power with a Theveninsource, transmission lines, and no isolators at the source. The Theveninsource 210 can include an ideal vector voltage source v_(s) incombination with a series impedance Z_(S) _(_) _(Thev), where Z_(S) _(_)_(Thev) is a diagonal matrix with diagonal elements equal to the seriesimpedance Z_(S) _(_) _(Thev1) and Z_(S) _(_) _(Thev2) for each voltagesource v_(s1) and v_(s2), respectively. The impedance looking into atransmission line can be given by

Z _(in)(l)=Z ₀(Z+jZ ₀ I ₂ tan(2πl)(Z ₀ I ₂ +jZ tan(2πl))⁻¹,

where Z₀ can be the impedance of the transmission line, l can be thelength of the transmission line in wavelengths, and Z can be theimpedance matrix for the combination of the antenna array and anyimpedance matching circuitry between the transmission line and theantenna array.

The transmit power can be given by

Re(v _(S) ^(H)(Z _(S) _(_) _(Thev) +Z _(in)(l))^(−H) Z _(in)(l)(Z _(S)_(_) _(Thev) +Z _(in)(l))⁻¹ v _(S))=v _(S) ^(H)(Z _(S) _(_) _(Thev) +Z_(in)(l))^(−H) Re(Z _(in)(l))(Z _(S) _(_) _(Thev) +Z _(in)(l))⁻¹ v _(S),

where the matrix Z_(S) _(_) _(Thev) can be a diagonal matrix withelements equal to the source impedances in series with the Theveninvoltage sources. This expression can be further simplified as thequadratic form

v _(S) ^(H)(Z _(S) _(_) _(Thev) +Z _(in)(l))^(−H) Re(Z _(in)(l))(Z _(S)_(_) _(Thev) +Z _(in)(l))⁻¹ v _(S) =v _(S) ^(H) Q _(Thev)(Z _(S) _(_)_(Thev) ,Z _(in)(l))v _(S)

where

Q _(Thev)(Z _(S) _(_) _(Thev) ,Z _(in)(l))=(Z _(S) _(_) _(Thev) +Z_(in)(l))^(−H) Re(Z _(in)(l))(Z _(S) _(_) _(Thev) +Z _(in)(l))⁻¹.

As an example of a two-element array of half-wavelength dipoles withhalf-wavelength spacing, the impedance matrix for this array can begiven by

$Z = {\begin{bmatrix}{73 + {j \cdot 42.5}} & {{- 13} - {j \cdot 25}} \\{{- 13} - {j \cdot 25}} & {73 + {j \cdot 42.5}}\end{bmatrix}.}$

The additional following parameters for this example can be assumed as:source impedance=50 ohms; transmission line impedance=50 ohms; andtransmission line length=one-quarter wavelength.

A voltage source of the form v(θ)=[1 exp (jθ)]^(T) can be considered forwhich the L2 norm of the precoder v(θ) can be independent of the phaseθ, so that

∥v(θ)∥²=2

for all θ.

FIG. 5 is an example illustration 500 of variation of transmit power asa function of phase θ for a two-element dipole array withhalf-wavelength spacing driven by a 50 ohm Thevenin source and 50 ohmtransmission lines according to a possible embodiment. It can be notedthat the transmit power varies by 1.3 dB even though the L2 norm of theprecoder is held constant.

FIG. 6 is an example illustration 600 of a Norton source and two-portantenna model with a transmission line according to a possibleembodiment. FIG. 7 is an example illustration 700 of a matching networkexplicitly shown between a Norton source and an antenna array withimpedance matrix Z according to a possible embodiment. FIG. 8 is anexample illustration 800 of a matching network combined with an antennaarray with a resulting combined impedance matrix Z′ according to apossible embodiment, where the combined impedance matrix Z′ can be usedinstead of the impedance matrix Z in equations herein to account for amatching network.

According to this embodiment, the transmit power with a Norton source610, transmission lines 620, an antenna array 630, and no isolators atthe source 610, the impedance looking into a transmission line can begiven by

Z _(in)(l)=Z ₀(Z+jZ ₀ I ₂ tan(2πl))(Z ₀ I ₂ +jZ tan(2πl))⁻¹,

where Z₀ can be the impedance of the transmission line, l can be thelength of the transmission line in wavelengths, and Z can be theimpedance matrix for the combination of the antenna array and anyimpedance matching circuitry between the transmission line and anantenna array.

The transmit power can be given by

Re(i _(S) ^(H) Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z_(in)(l))^(−H) Z _(in)(l)(Z _(S) _(_) _(Nor) +Z _(in)(l))⁻¹ Z _(S) _(_)_(Nor))i _(S) =i _(S) ^(H)(Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z_(in)(l))^(−H) Re(Z _(in)(l))(Z _(S) _(_) _(Nor) +Z _(in)(l))⁻¹ Z _(S)_(_) _(Nor))i _(S),

where the matrix Z_(S) _(_) _(Nor) can be a diagonal matrix withelements equal to the shunt source impedances in parallel with theNorton current sources. This expression can be further simplified as thequadratic form

i _(S) ^(H)(Z _(S) _(_) _(Nor) ^(H)(Z _(S) _(_) _(Nor) +Z _(in)(l))^(−H)Re(Z _(in)(l))(Z _(S) _(_) _(Nor) +Z _(in)(l))⁻¹ Z _(S) _(_) _(Nor))^(i)_(S) =i _(S) ^(H) Q _(Nor)(Z _(S) _(_) _(Nor) ,Z _(in)(l))i _(S)

where

Q _(Nor)(Z _(S) _(_) _(Nor) ,Z _(in)(l))=Z _(S) _(_) _(Nor) ^(H)(Z _(S)_(_) _(Nor) +Z _(in)(l))^(−H) Re(Z _(in)(l))(Z _(S) _(_) _(Nor) +Z_(in)(l))⁻¹ Z _(S) _(_) _(Nor).

According to a possible embodiment for transmit power with a Theveninsource, transmission lines, and isolators at the source, when theisolator is used at the source, the transmitter does not see the voltageand current reflected from the antenna array (and any matchingcircuitry). Instead, the reflected voltage and current are routed awayfrom the transmitter and into a matched load. As a result, the impedancelooking into the transmission line can just be the transmission lineimpedance Z₀ and thus may not depend on the impedance of the antennaarray.

For a Thevenin source, the forward voltage wave into the transmissionline can be given by

V ⁺ =Z ₀ Z _(S) ⁻¹ v _(S).

At the antenna array load, the reflected voltage wave can be given by

$\begin{matrix}{V^{-} = {\left( {Z + {Z_{0}I_{2}}} \right)^{- 1}\left( {Z - {Z_{0}I_{2}}} \right)V^{+}}} \\{= {SV}^{+}}\end{matrix},$

where S can be the scattering matrix given by

S=(Z+Z ₀ I ₂)⁻¹(Z−Z ₀ I ₂).

The total voltage at the load can be given by

$V_{tot} = {\begin{matrix}{{V^{+} + V^{-}} = {\left( {I_{2} + S} \right)V^{+}}} \\{= {{Z_{0}\left( {I_{2} + S} \right)}Z_{S\_ Thev}^{- 1}v_{S}}}\end{matrix}.}$

The total current at the load can be given by

$I_{tot} = {\begin{matrix}{{I^{+} + I^{-}} = {Z_{0}^{- 1}\left( {V^{+} - V^{-}} \right)}} \\{= {{Z_{0}^{- 1}\left( {I_{2} - S} \right)}V^{+}}} \\{= {{Z_{0}^{- 1}\left( {I_{2} - S} \right)}Z_{0}Z_{S\_ Thev}v_{S}}} \\{= {\left( {I_{2} - S} \right)Z_{S\_ Thev}v_{S}}}\end{matrix}.}$

The power delivered to the load can then be given by the quadratic form

$\begin{matrix}{{{Re}\left( {V_{tot}^{H}I_{tot}} \right)} = {Z_{0}{{Re}\left( {v_{S}^{H}{Z_{S\; \_ \; {Thev}}^{- H}\left( {I_{2} + S} \right)}^{H}\left( {I_{2} - S} \right)Z_{S\; \_ \; {Thev}}^{- 1}v_{S\;}} \right)}}} \\{= {Z_{0}{{Re}\left( {v_{S}^{H}{Z_{S\; \_ \; {Thev}}^{- H}\left( {I_{2} + {S^{H}S} - {2\; {{Im}(S)}}} \right)}Z_{S\; \_ \; {Thev}}^{- 1}v_{S\;}} \right)}}} \\{= {Z_{0}v_{S}^{H}{Z_{S\; \_ \; {Thev}}^{- H}\left( {I_{2} - {{Re}\left( {S^{H}S} \right)}} \right)}Z_{S\; \_ \; {ThevS}}^{- H}{v.}}}\end{matrix}$

In the event that the source impedances are equal, this expression forthe transmitted power can be expressed as

${{{Re}\left( {V_{tot}^{H}I_{tot}} \right)} = {\frac{Z_{0}}{{Z_{S\; \_ \; {Thev}}}^{2}}{v_{S}^{H}\left( {I_{2} - {{Re}\left( {S^{H}S} \right)}} \right)}v}},$

or more simply as

Re(V_(tot)^(H)I_(tot)) = v_(S)^(H)Q_(Thev _ iso)(Z₀, Z_(S _ Thev), S)v, where${Q_{{Thev}\; \_ \; {iso}}\left( {Z_{0},Z_{S\; \_ \; {Thev}},S} \right)}\frac{Z_{0}}{{Z_{S\; \_ \; {Thev}}}^{2}}{\left( {I_{2} - {{Re}\left( {S^{H}S} \right)}} \right).}$

FIG. 9 is an example graph 900 of variation of transmit power as afunction of phase for a two-element dipole array with half-wavelengthspacing driven by a 50 ohm Thevenin source with isolators and 50 ohmtransmission lines according to a possible embodiment. The same exampleas previously given of a two-element array of half-wavelength dipoleswith half-wavelength spacing can be considered. The impedance matrix forthis array can be given by

$Z = {\begin{bmatrix}{73 + {j \cdot 42.5}} & {{- 13} - {j \cdot 25}} \\{{- 13} - {j \cdot 25}} & {73 + {j \cdot 42.5}}\end{bmatrix}.}$

As before, the transmission line impedance can be assumed as 50 ohms anda voltage source of the form v(θ)==[1 exp (jθ)]^(T) for which the L2norm of the precoder v(θ) is independent of the phase θ can beconsidered so that

∥v(θ)∥²=2

for all θ.

For this example, the variation of the transmit power as a function ofthe phase is shown in the graph 900. It can be noted that the transmitpower varies by 1.3 dB even though the L2 norm of the precoder is heldconstant.

According to a possible embodiment for transmit power with a Nortonsource, transmission lines, and isolators at the source, when anisolator is used at the source, the transmitter does not see the voltageand current reflected from the antenna array (and any matchingcircuitry). Instead, the reflected voltage and current are routed awayfrom the transmitter and into a matched load. As a result, the impedancelooking into the transmission line can just be the transmission lineimpedance Z₀ and thus may not depend on the impedance of the antennaarray.

For a Norton source, the forward voltage wave into the transmission linecan be given by

V ⁺=(Z ₀ I ₂ +Z _(S) _(_) _(Nor))⁻¹ Z ₀ Z _(S) _(_) _(Nor) i _(S).

As in the previous case, the antenna array load, the reflected voltagewave can be given by

$\begin{matrix}{V^{-} = {\left( {Z + {Z_{0}I_{2}}} \right)^{- 1}\left( {Z - {Z_{0}I_{2}}} \right)V^{+}}} \\{{= {SV}^{+}},}\end{matrix}$

where S can be the scattering matrix given by

S=(Z+Z ₀ I ₂)⁻¹(Z−Z ₀ I ₂).

The total voltage at the load can be given by

$\begin{matrix}{V_{tot} = {V^{+} + V^{-}}} \\{= {\left( {I_{2} + S} \right)V^{+}}} \\{= {{Z_{0}\left( {I_{2} + S} \right)}\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)^{- 1}Z_{S\; \_ \; {Nor}}{i_{S}.}}}\end{matrix}$

The total current at the load can be given by

$\begin{matrix}{I_{tot} = {I^{+} + I^{-}}} \\{= {Z_{0}^{- 1}\left( {V^{+} - V^{-}} \right)}} \\{= {{Z_{0}^{- 1}\left( {I_{2} - S} \right)}V^{+}}} \\{= {{Z_{0}^{- 1}\left( {I_{2} - S} \right)}\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)^{- 1}Z_{0}Z_{S\; \_ \; {Nor}}i_{S}}} \\{= {\left( {I_{2} - S} \right)\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)^{- 1}Z_{S\; \_ \; {Nor}}{i_{S}.}}}\end{matrix}$

The power delivered to the load can then be given by the quadratic form

$\begin{matrix}{{{Re}\left( {V_{tot}^{H}I_{tot}} \right)} = {Z_{0}{{Re}\left( {i_{S}^{H}{Z_{S\; \_ \; {Nor}}^{H}\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)}^{- H}\left( {I_{2} + S} \right)^{H}\left( {I_{2} - S} \right)} \right.}}} \\\left. {\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)^{- 1}Z_{S\; \_ \; {Nor}}i_{S}} \right) \\{= {Z_{0}{{Re}\left( {i_{S}^{H}{Z_{S\; \_ \; {Nor}}^{H}\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)}^{- H}\left( {I_{2} - {S^{H}S} - {2{{Im}(S)}}} \right)} \right.}}} \\\left. {\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)^{- 1}Z_{S\; \_ \; {Nor}}i_{S}} \right) \\{= {Z_{0}i_{S}^{H}{Z_{S\; \_ \; {Nor}}^{H}\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)}^{- H}\left( {I_{2} - {{Re}\left( {S^{H}S} \right)}} \right)}} \\{\left. {\left( {{Z_{0}I_{2}} + Z_{S\; \_ \; {Nor}}} \right)^{- 1}Z_{S\; \_ \; {Nor}}i_{S}} \right).}\end{matrix}$

In the event that the source impedances are equal, this expression forthe transmitted power can be expressed as

${{Re}\left( {V_{tot}^{H}I_{tot}} \right)} = {\frac{Z_{0}{Z_{S\; \_ \; {Nor}}}^{2}}{{{Z_{0} + Z_{S\; \_ \; {Nor}}}}^{2}}{i_{S}^{H}\left( {I_{2} - {{Re}\left( {S^{H}S} \right)}} \right)}i}$

or more simply as

Re(V_(tot)^(H)I_(tot)) = i_(S)^(H)Q_(Nor _ iso)(Z₀, Z_(S _ Nor), S)i, where${Q_{{Nor}\; \_ \; {iso}}\left( {Z_{0},Z_{S\; \_ \; {Nor}},S} \right)}\frac{Z_{0}{Z_{S\; \_ \; {Nor}}}^{2}}{{{Z_{0} + Z_{S\; \_ \; {Nor}}}}^{2}}{\left( {I_{2} - {{Re}\left( {S^{H}S} \right)}} \right).}$

According to a possible embodiment for power variation of precoders withequal Euclidean norm, in the four cases above, the transmit power can beexpressed as a quadratic form with respect to the currents or voltagesused to drive the antenna arrays. It is apparent that even if theEuclidean norm of the precoder is held constant, the transmit power canvary as a function of the relative phases of the input voltages or inputcurrents. As a result, the implicit assumption used in the design of the3GPP LTE and IEEE 802.16 codebooks—that precoders with equal Euclideannorm map to antenna patterns with equal transmit power—is incorrect whenthe antennas are coupled and transmission lines are used between thetransmitter and the antenna array.

Two methods can be used for mapping precoders with equal Euclidean normto antenna patterns with equal power. In a first method, a separatereal-valued scaling can be applied to each precoder so that the transmitpowers are equalized. For example, for the case in which a Theveninsource is used to drive the array and an isolator is used at thetransmitter, the transmit power can be given by

Re(V _(tot) ^(H) I _(tot))=v _(S) ^(H) Q _(Thev) _(_) _(iso)(Z ₀ ,Z _(S)_(_) _(Thev) ,S)v.

In order to equalize the transmit power for precoders having equalEuclidean norm, a precoder v_(k) can be normalized into a normalizedprecoder v_(k,norm) by defining v_(k,norm) as

${v_{k,{norm}} = \frac{\alpha \; v_{k}}{\left( {v_{k}^{H}{Q_{{Thev}\; \_ \; {iso}}\left( {Z_{0},Z_{S\; \_ \; {Thev}},S} \right)}v_{k}} \right)^{1/2}}},$

where α is a constant that is the same for all precoders. It should benoted that some equivalent variables in equivalent equations throughoutthis disclosure may be different for ease of description in thecorresponding sections. In this method, the receiver should know thescaling factor applied to each precoder. In particular, the scalingfactor for each precoder should be signaled to the receiver. So, for thek-th precoder, the scaling factor

$\frac{\alpha \; v_{k}}{\left( {v_{k}^{H}{Q_{{Thev}\; \_ \; {iso}}\left( {Z_{0},Z_{S\; \_ \; {Thev}},S} \right)}v_{k}} \right)^{1/2}}$

can be sent to the receiver. There are other ways to signal the scalinginformation to the receiver. For example, the parameter α and the matrix

Q _(Thev) _(_) _(iso)(Z ₀ ,Z _(S) _(_) _(Thev) ,S)

can be signaled to the receiver. Since the matrix Q_(Thev) _(_) _(iso)can be Hermitian, the matrix coefficients can be signaled to thereceiver in the form of Mx(M−1)/2 complex values and M real values,where M can be the number of transmit antennas.

For a second method, a transformation can be performed on the set ofequal Euclidean norm precoders so that the transformed precoders can allmap to antenna patterns with equal power. For this reason, the matrixQ_(Thev) _(_) _(iso) can be factored as

Q _(Thev) _(_) _(iso)(Z ₀ ,Z _(S) _(_) _(Thev) ,S)=P P ^(H),

where this factorization can be non-unique. One factorization of thistype is the Cholesky factorization where the matrix P can be lowertriangular (and thus P^(H) can be upper triangular). Otherfactorizations having this form can be generated by noting that becauseQ_(Thev) _(_) _(iso) can be Hermitian, the eigendecomposition ofQ_(Thev) _(_) _(iso) can have the form

Q _(Thev) _(_) _(iso) =UΛU ^(H),

where the columns of U can be the eigenvectors of Q_(Thev) _(_) _(iso)and the matrix Λ can be diagonal. The diagonal elements of Λ can be theeigenvalues corresponding to the eigenvectors of Q_(Thev) _(_) _(iso),where the eigenvalues in Λ can be in the same order as the correspondingeigenvectors in U. Using this eigendecomposition, the following can bedefined:

P=UΛ ^(1/2),

where Λ^(1/2) is the square root of the matrix A. It can be noted thatthe eigendecomposition of the matrix Q_(Thev) _(_) _(iso) may not beunique since the eigenvectors forming the columns of U can be placed inany order. If the dimension of Q_(Thev) is M×M, then Q_(Thev) _(_)_(iso) has M eigenvectors and there are M factorial (M!=M*(M−1)*(M−2)* .. . *1) possible orderings of these eigenvectors. Also, given the matrixU and the diagonal matrix of the corresponding eigenvalues Λ, the squareroot matix Λ^(1/2) can be non-unique since each eigenvalue has both apositive and a negative square root (all of the eigenvalues of Q_(Thev)_(_) _(iso) are non-negative). Thus, given the matrix Λ, there can be2^(M) possible matrices Λ^(1/2). However, given the matrix A there isonly one matrix Λ^(1/2) for which all of the values are non-negative andthis can be referred to as the positive square root.

For the remainder of this section, the following definition can be used:

P=UΛ ^(1/2),

where Λ^(1/2) can be the positive square root of the matrix A. Theordering of the eigenvectors of Q_(Thev) _(_) _(iso) within the columnsof U may not matter, though the ordering of the eigenvalues in Λ shouldcorrespond to the ordering of the eigenvectors in U. It can be notedthat because the eigenvectors are orthonormal, it follows that

P ^(−H) =UΛ ^(−1/2).

Now define

$\begin{matrix}{v_{s} = {P^{- H}w}} \\{{= {U\; \Lambda^{{- 1}/2}w}},}\end{matrix}$

so that v_(S) is the sum of the projections of w onto the eigenvectorsof Q_(Thev) _(_) _(iso) scaled by the inverse square root of thecorresponding eigenvalues. Note that

$\begin{matrix}{{v_{S}^{H}Q_{Thev}v_{S\;}} = {w^{H}P^{- 1}Q_{{Thev}\; \_ \; {iso}}P^{- H}w}} \\{= {w^{H}P^{- 1}{PP}^{H}P^{- H}w}} \\{= {w^{H}w}} \\{= {{w}_{2}^{2}.}}\end{matrix}$

So long as the reference symbol precoders use the same transformation asthe data symbol precoders, the receiver can use the existing precodersto estimate the channel, and the receiver does not need to know theprecoder transformation that was used at the transmitter. Thus, if eachprecoder w is transformed into a voltage vector v_(S) using thetransformation v_(S)=P_(Thev) ^(−H)w, all of the precoders can map toequal energy patterns so long as all of the precoders have the same L²norm.

Methods described for mapping equal Euclidean norm precoders to antennapatterns with equal power can be used for all four of the cases givenabove in which the transmit power is quadratic with respect to thevector voltage (for a Thevenin source) or vector current (for a Nortonsource). Therefore, embodiments can be applied at least for thefollowing four cases considered in this description: transmit power witha Thevenin source, transmission lines, and no isolators at the source;transmit power with a Norton source, transmission lines, and noisolators at the source; transmit power with a Thevenin source,transmission lines, and isolators at the source; and transmit power witha Norton source, transmission lines, and isolators at the source.

FIG. 10 is an example flowchart 1000 illustrating the operation of awireless communication device, such as the device 110 and/or the device120, according to a possible embodiment. For example, the method of theflowchart 1000 can be used in cases where transmission lines are usedwithout isolators, in cases where isolators are used at a source of thesignal for transmission along with transmission lines, and other cases.At 1010, a plurality of precoders can be received from a codebook in atransmitter having an antenna array.

At 1020, each precoder of the plurality of precoders can be transformedto a transformed precoder such that the transmit power for eachtransformed precoder is equal to the transmit power for each of othertransformed precoders of the plurality of precoders. The transmit powercan be expressed as a quadratic form with respect to the correspondingprecoder. The quadratic form can be non-negative definite. The quadraticform can be based on a transmission line impedance of a transmissionline between a signal source and the antenna array. The quadratic formcan also be based on an impedance matrix of the antenna array. This canbe used both when isolators are used at a source of the signal fortransmission and also when isolators are not used. The quadratic formcan also be based on a matching network between the signal source andthe antenna array. A matching network can transform the impedance matrixof the antenna array to improve the power transfer between the sourceand the antenna array. The quadratic form can further be based on animpedance of the antenna array, a signal source impedance, and length ofthe transmission line. For example, the quadratic form may be based onthe length of the transmission line when isolators are not used at thesignal source. Additionally, the quadratic form can be a function of thetransmission line impedance of a transmission line between the signalsource and the antenna array, an impedance of a source of the signal fortransmission, an impedance matching network, a scattering matrix, andother information, where the quadratic form can be a function of some orall of the information.

The transformation can equalize the transmit power for all of theplurality of precoders. Transforming can also transform the precoders toequal energy precoders when transmission lines are used between a sourceof the signal and the antenna array. Furthermore, both data symbolprecoders and reference symbol precoders can be transformed by the sametransformation.

Each precoder can be transformed by scaling each precoder by an inversesquare root of a transmit power that results from the correspondingprecoder before scaling is applied. Scaling can include normalizing aprecoder into a normalized precoder based on the quadratic form. Forexample, scaling can include normalizing a precoder w_(k), into anormalized precoder v_(S,norm) based on

${v_{S,{norm}} = \frac{\alpha \; w_{k}}{\left( {w_{k}^{H}{Q_{{Thev}\; \_ \; {iso}}\left( {Z_{0},Z_{S\; \_ \; {Thev}},S} \right)}w_{k}} \right)^{1/2}}},$

where α can be a constant that is the same for all precoders andQ_(Thev) _(_) _(iso) can be a matrix that can be a function of ascattering matrix S, an impedance Z_(S) _(_) _(Thev) of a source of thesignal for transmission, and a transmission line impedance Z₀ of thetransmission line. The scattering matrix S can be a function of theimpedance matrix of the antenna array. This equation can cover the casein which a Thevenin source model is used with isolators. Other similarequations can cover other cases, such when a Norton source model isused, when an isolator is not used, and other cases.

Scattering parameters may or may not be used for the case in which noisolators are used at the source. Scattering parameters can depend onthe transmission line impedance, the antenna impedance, and any matchingcircuitry. If isolators are used at the source, the quadratic form forthe transmit power can be based on only the scattering parameterswithout individual knowledge of the transmission line impedance and theimpedance matrix of the antenna array. In the case with no isolators atthe source, the transmission line impedance and the antenna arrayimpedance can be used separately. In the case that isolators are used atthe source, a power variation may only depend on a scattering matrixwithout the need for the additional knowledge of the transmission lineimpedance and the antenna array impedance separately.

Also or alternately, each precoder can be transformed by multiplyingeach precoder by a transformation matrix such that the resulting set ofprecoders each map to antenna patterns having the same power. Thetransformation matrix can be based on an impedance matrix of the antennaarray seen at the signal source as a function of the transmission linelength, the transmission line impedance, and an impedance of the antennaarray. The transmission line length can be measured in wavelengths. Thetransformation matrix can also be based on a diagonal matrix oftransmitter source impedances. As an example for a Thevenin sourcemodel, the transmitter can include a transmitter source of the signalfor transmission and the transformation can be based on

v _(S) =P _(Thev) ^(−H) w,

where w can be a precoder from a set of precoders, v_(S) can be thetransformed precoder in terms of voltage, and P_(Thev) can be based on

Q _(Thev) =P _(Thev) P _(Thev) ^(H),

where

Q _(Thev)=(Z _(S) _(_) _(Thev) +Z _(in)(l))=(Z _(S) _(_) _(Thev) +Z_(in)(l))^(−H) Re(Z _(in)(l))(Z _(S) _(_) _(Thev) +Z _(in)(l))⁻¹,

where l can be a transmission line length measured in wavelengths, Z_(S)_(_) _(Thev) can be a diagonal matrix of transmitter source impedances,and Z_(in) can be the impedance matrix of the antenna array seen at thesource as a function of the transmission line length, the transmissionline impedance, and an impedance of the antenna array. v_(S) can be atwo element vector including two source voltages v_(s1) and v_(s2). Thisequation can be used for a Thevenin source model where another similarequation can be used for a Norton source model.

As an example for a Norton source model, the transmitter can include atransmitter source of the signal for transmission and the transformationcan be based on

i _(S) =P _(Not) ^(−H) w,

where w can be a precoder from a set of precoders, i can be thetransformed precoder, and P_(Nor) can be based on

Q _(Not) =P _(Not) P _(Not) ^(H),

where

Q _(Nor)(Z _(S) _(_) _(Nor) ,Z _(in)(l))=Z _(S) _(_) _(Not) ^(H)(Z _(S)_(_) _(Nor) ±Z _(in)(l))^(−H) Re(Z _(in)(l))(Z _(S) _(_) _(Nor) +Z_(in)(l))⁻¹ Z _(S) _(_) _(Nor),

where l can be a transmission line length measured in wavelengths, Z_(S)_(_) _(Nor) can be a diagonal matrix of transmitter source impedances,and Z_(in) can be the impedance matrix of the antenna array seen at thesource as a function of the transmission line length, the transmissionline impedance, and an impedance of the antenna array.

At 1030, a signal can be received from the signal source. At 1040, atransformed precoder of the plurality of transformed precoders can beapplied to the signal to generate a precoded signal for transmissionover a physical channel. At 1050, the precoded signal can betransmitted. At 1060, a scaling factor used for the scaling can betransmitted.

FIG. 11 is an example flowchart 1100 illustrating the operation of awireless communication device, such as the device 110 and/or the device120, according to a possible embodiment. At 1110, a precoded signalincluding reference symbols can be received. At 1120, channels for thereference symbols can be estimated. At 1130, a channel for the datasymbols can be estimated by taking an inner product of a conjugate of adata symbol precoder and the reference symbol channel estimates. Thechannel estimate for the reference symbols and the data symbols can beperformed when the antenna array is used to receive precoded signals.For example, when a device including the antenna array is transmitting,it can transmit precoded signals and when the device is receiving, itcan receive precoded signals. At 1140, received data symbols can bedemodulated based on the estimated channel

It should be understood that, notwithstanding the particular steps asshown in the figures, a variety of additional or different steps can beperformed depending upon the embodiment, and one or more of theparticular steps can be rearranged, repeated or eliminated entirelydepending upon the embodiment. Also, some of the steps performed can berepeated on an ongoing or continuous basis simultaneously while othersteps are performed. Furthermore, different steps can be performed bydifferent elements or in a single element of the disclosed embodiments.

FIG. 12 is an example block diagram of an apparatus 1200, such as thetransmitting device 110, according to a possible embodiment. Theapparatus 1200 can be a base station, a UE, or any other transmittingand/or receiving apparatus. The apparatus 1200 can include a housing1210, a controller 1220 coupled to the housing 1210, audio input andoutput circuitry 1230 coupled to the controller 1220, a display 1240coupled to the controller 1220, a transceiver 1250 coupled to thecontroller 1220, an antenna array including plurality of antennas, suchas antennas 1255 and 1257, coupled to the transceiver 1250, a userinterface 1260 coupled to the controller 1220, a memory 1270 coupled tothe controller 1220, and a network interface 1280 coupled to thecontroller 1220. The apparatus 1200 can perform the methods described inall the embodiments.

The display 1240 can be a viewfinder, a liquid crystal display (LCD), alight emitting diode (LED) display, a plasma display, a projectiondisplay, a touch screen, or any other device that displays information.The transceiver 1250 can include a transmitter and/or a receiver. Thetransceiver 1250 can also include a signal source or the signal sourcecan be located elsewhere on the apparatus 1200. The plurality ofantennas 1255 and 1257 can be an antenna array. A transmission line (notshown) can be coupled between the signal source 1252 and the antennaarray. The transmission line can have a transmission line impedance. Theplurality of antennas 1255 and 1257 can include two or more antennas.The antennas 1255 and 1257 can be mutually coupled in that voltage orcurrent applied to one antenna element induces a voltage or current onanother antenna element in the antenna array. The audio input and outputcircuitry 1230 can include a microphone, a speaker, a transducer, or anyother audio input and output circuitry. The user interface 1260 caninclude a keypad, a keyboard, buttons, a touch pad, a joystick, a touchscreen display, another additional display, or any other device usefulfor providing an interface between a user and an electronic device. Thenetwork interface 1280 can be a Universal Serial Bus (USB) port, anEthernet port, an infrared transmitter/receiver, an IEEE 1398 port, aWLAN transceiver, or any other interface that can connect an apparatusto a network, device, or computer and that can transmit and receive datacommunication signals. The memory 1270 can include a random accessmemory, a read only memory, an optical memory, a flash memory, aremovable memory, a hard drive, a cache, or any other memory that can becoupled to a wireless communication device.

The apparatus 1200 or the controller 1220 may implement any operatingsystem, such as Microsoft Windows®, UNIX®, or LINUX®, Android™, or anyother operating system. Apparatus operation software may be written inany programming language, such as C, C++, Java or Visual Basic, forexample. Apparatus software may also run on an application framework,such as, for example, a Java® framework, a .NET® framework, or any otherapplication framework. The software and/or the operating system may bestored in the memory 1270 or elsewhere on the apparatus 1200. Theapparatus 1200 or the controller 1220 may also use hardware to implementdisclosed operations. For example, the controller 1220 may be anyprogrammable processor. Disclosed embodiments may also be implemented ona general-purpose or a special purpose computer, a programmedmicroprocessor or microprocessor, peripheral integrated circuitelements, an application-specific integrated circuit or other integratedcircuits, hardware/electronic logic circuits, such as a discrete elementcircuit, a programmable logic device, such as a programmable logicarray, field programmable gate-array, or the like. In general, thecontroller 1220 may be any controller or processor device or devicescapable of operating a wireless communication device and implementingthe disclosed embodiments. While the controller 1220 is illustrated asone block and operations of the controller 1220 can be performed in oneelement, the controller 1220 can alternately be distributed betweendifferent elements of the apparatus 1200 as well as distributed throughcloud computing. For example, different controllers can exist on theapparatus 1200 to perform different operations for different elements onthe apparatus 1200, a master controller can perform all of theoperations of the apparatus 1200, and/or a master controller can performsome overall operations and distributed controllers can perform otheroperations for other elements on the apparatus 1200.

In operation, the memory 1270 can store a codebook including a pluralityof precoders. The controller 1220 can receive a plurality of precodersfrom the codebook in the memory 1270.

The controller 1220 can transform each precoder of the plurality ofprecoders to a transformed precoder such that the transmit power foreach transformed precoder is equal to the transmit power for each of theother transformed precoders of the plurality of precoders. The transmitpower can be expressed as a quadratic form with respect to thecorresponding precoder. The quadratic form can be based on thetransmission line impedance. The quadratic form can also be based on animpedance matrix of the antenna array, a matching network between thesignal source and the antenna array, an impedance of a transmissionline, a signal source impedance, and/or a length of the transmissionline.

Each precoder can be transformed by scaling each precoder by an inversesquare root of a transmit power that results from the correspondingprecoder before scaling is applied. Scaling can include normalizing aprecoder into a normalized precoder based on the quadratic form. Also oralternately, each precoder can be transformed by multiplying eachprecoder by a transformation matrix such that the resulting set ofprecoders each map to antenna patterns having the same power. Thetransformation matrix can be based on an impedance matrix of the antennaarray seen at the signal source as a function of the transmission linelength, the transmission line impedance, and an impedance of the antennaarray.

The controller 1220 can receive a signal from the signal source 1252.The controller 1220 can apply a transformed precoder of the plurality oftransformed precoders to the signal to generate a precoded signal fortransmission over a physical channel. The transceiver 1250 can transmitthe precoded signal.

The method of this disclosure can be implemented on a programmedprocessor. However, the controllers, flowcharts, and modules may also beimplemented on a general purpose or special purpose computer, aprogrammed microprocessor or microcontroller and peripheral integratedcircuit elements, an integrated circuit, a hardware electronic or logiccircuit such as a discrete element circuit, a programmable logic device,or the like. In general, any device on which resides a finite statemachine capable of implementing the flowcharts shown in the figures maybe used to implement the processor functions of this disclosure.

While this disclosure has been described with specific embodimentsthereof, it is evident that many alternatives, modifications, andvariations will be apparent to those skilled in the art. For example,various components of the embodiments may be interchanged, added, orsubstituted in the other embodiments. Also, all of the elements of eachfigure are not necessary for operation of the disclosed embodiments. Forexample, one of ordinary skill in the art of the disclosed embodimentswould be enabled to make and use the teachings of the disclosure bysimply employing the elements of the independent claims. Accordingly,embodiments of the disclosure as set forth herein are intended to beillustrative, not limiting. Various changes may be made withoutdeparting from the spirit and scope of the disclosure.

In this document, relational terms such as “first,” “second,” and thelike may be used solely to distinguish one entity or action from anotherentity or action without necessarily requiring or implying any actualsuch relationship or order between such entities or actions. The phrase“at least one of,” “at least one selected from the group of,” or “atleast one selected from” followed by a list is defined to mean one,some, or all, but not necessarily all of, the elements in the list. Theterms “comprises,” “comprising,” “including,” or any other variationthereof, are intended to cover a non-exclusive inclusion, such that aprocess, method, article, or apparatus that comprises a list of elementsdoes not include only those elements but may include other elements notexpressly listed or inherent to such process, method, article, orapparatus. An element proceeded by “a,” “an,” or the like does not,without more constraints, preclude the existence of additional identicalelements in the process, method, article, or apparatus that comprisesthe element. Also, the term “another” is defined as at least a second ormore. The terms “including,” “having,” and the like, as used herein, aredefined as “comprising.” Furthermore, the background section is writtenas the inventor's own understanding of the context of some embodimentsat the time of filing and includes the inventor's own recognition of anyproblems with existing technologies and/or problems experienced in theinventor's own work.

We claim:
 1. A method comprising: receiving a plurality of precodersfrom a codebook in a transmitter having an antenna array; transformingeach precoder of the plurality of precoders to a transformed precodersuch that the transmit power for each transformed precoder is equal tothe transmit power for each of other transformed precoders of theplurality of precoders, where the transmit power is expressed as aquadratic form with respect to the corresponding precoder, where thequadratic form is based on a transmission line impedance of atransmission line between a signal source and the antenna array;receiving a signal from the signal source; applying a transformedprecoder of the plurality of transformed precoders to the signal togenerate a precoded signal for transmission over a physical channel; andtransmitting the precoded signal.
 2. The method according to claim 1,wherein the quadratic form is also based on an impedance matrix of theantenna array.
 3. The method according to claim 1, wherein the quadraticform is also based on a matching network between the signal source andthe antenna array.
 4. The method according to claim 1, wherein thequadratic form is also based on an impedance of the antenna array, asignal source impedance, and length of the transmission line.
 5. Themethod according to claim 1, wherein each precoder is transformed byscaling each precoder by an inverse square root of a transmit power thatresults from the corresponding precoder before scaling is applied. 6.The method according to claim 5, wherein scaling comprises normalizing aprecoder into a normalized precoder based on the quadratic form.
 7. Themethod according to claim 5, further comprising transmitting a scalingfactor used for the scaling.
 8. The method according to claim 1, whereineach precoder is transformed by multiplying each precoder by atransformation matrix such that the resulting set of precoders each mapto antenna patterns having the same power.
 9. The method according toclaim 8, wherein the transformation matrix is based on an impedancematrix of the antenna array seen at the signal source as a function ofthe transmission line length, the transmission line impedance, and animpedance of the antenna array.
 10. The method according to claim 1,wherein the transformation equalizes the transmit power for all of theplurality of precoders.
 11. The method according to claim 1, whereintransforming transforms the precoders to equal energy precoders whentransmission lines are used between a source of the signal and theantenna array.
 12. The method according to claim 1, wherein both datasymbol precoders and reference symbol precoders are transformed by thesame transformation.
 13. The method according to claim 12, furthercomprising: receiving a precoded signal including reference symbols;estimating channels for the reference symbols; estimating a channel forthe data symbols by taking an inner product of a conjugate of a datasymbol precoder and the reference symbol channel estimates; anddemodulating received data symbols based on the estimated channel. 14.An apparatus comprising: an antenna array; a transceiver coupled to theantenna array; a signal source coupled to the transceiver; atransmission line coupled between the signal source and the antennaarray, the transmission line having a transmission line impedance; amemory to store a codebook including a plurality of precoders; and acontroller coupled to the transceiver and the memory, the to receive aplurality of precoders from the codebook, transform each precoder of theplurality of precoders to a transformed precoder such that the transmitpower for each transformed precoder is equal to the transmit power foreach of other transformed precoders of the plurality of precoders, wherethe transmit power is expressed as a quadratic form with respect to thecorresponding precoder, where the quadratic form is based on thetransmission line impedance, receive a signal from the signal source,and apply a transformed precoder of the plurality of transformedprecoders to the signal to generate a precoded signal for transmissionover a physical channel, wherein the transceiver transmits the precodedsignal.
 15. The apparatus according to claim 15, wherein the quadraticform is also based on at least one selected from an impedance matrix ofthe antenna array, a matching network between the signal source and theantenna array, an impedance of the antenna array, a signal sourceimpedance, and length of the transmission line.
 16. The apparatusaccording to claim 14, wherein each precoder is transformed by scalingeach precoder by an inverse square root of a transmit power that resultsfrom the corresponding precoder before scaling is applied.
 17. Theapparatus according to claim 16, wherein scaling comprises normalizing aprecoder into a normalized precoder based on the quadratic form.
 18. Theapparatus according to claim 14, wherein each precoder is transformed bymultiplying each precoder by a transformation matrix such that theresulting set of precoders each map to antenna patterns having the samepower.
 19. The apparatus according to claim 18, wherein thetransformation matrix is based on an impedance matrix of the antennaarray seen at the signal source as a function of the transmission linelength, the transmission line impedance, and an impedance of the antennaarray.
 20. The apparatus according to claim 14, wherein the controllertransforms the precoders to equal energy precoders when transmissionlines are used between a source of the signal and the antenna array.